
Thomas Bellitto 


Dept. of Mathematics & Computer Science 
I am a postdoc at IMADA, University of Southern Denmark in Odense. I am a member of the project Packing, covering and partitions of digraphs directed by Jørgen BangJensen. My research involves graph theory, combinatorial optimisation, operations research, algorithmics, complexity theory and geometry. 
The chromatic number of a digraph D is the minimum number of colors needed to color the vertices of D such that each color class induces an acyclic subdigraph of D. A digraph is kcritical if its chromatic number is k but the chromatic number of all its proper subdigraphs is strictly smaller than k. We examine methods for creating infinite families of critical digraphs, the Dirac join and the directed and bidirected Hajós join. We prove that a digraph D has chromatic number at least k if and only if it contains a subdigraph that can be obtained from bidirected complete graphs on k vertices by (directed) Hajós joins and identifying nonadjacent vertices. Building upon that, we show that a digraph D has chromatic number at least k if and only if it can be constructed from bidirected complete graphs of k vertices by using directed and bidirected Hajós joins and identifying nonadjacent vertices (also called Ore joins), thereby transferring a wellknown result of Urquhart to digraphs. Finally, we prove a Gallaitype theorem that characterizes the structure of the low vertex subdigraph of a critical digraph, that is, the subdigraph, which is induced by the vertices that have indegree k−1 and outdegree k−1 in D.
We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.
This paper studies the problem of connecting edgecolouring. Given an undirected connected graph, our aim is to colour its edges with as few colours as possible so that there exists a properly coloured walk between every pair of vertices of the graph i.e. a walk that does not use consecutively two edges of the same colour. We establish that the problem can be solved in polynomial time in the size of the graph and we provide a characterization of the graphs that can be properly connected with k colours for every possible value of k.
The weak 2linkage problem for digraphs asks for a given digraph and vertices s1,s2,t1,t2 whether D contains a pair of arcdisjoint paths P1,P2 such that Pi is an (si,ti)path. This problem is NPcomplete for general digraphs but polynomially solvable for acyclic digraphs. Recently it was shown that if D is equipped with a weight function w on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak2linkage problem when both paths have to be shortest paths in D. In this paper we consider the unit weight case and prove that for every pair constants k1,k2, there is a polynomial algorithm which decides whether the input digraph D has a pair of arcdisjoint paths P1,P2 such that Pi is an (si,ti)path and the length of Pi is no more than d(si,ti)+ki, for i=1,2, where d(si,ti) denotes the length of the shortest (si,ti)path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution P1,P2 to the weak 2linkage problem where each path Pi has length at most d(si,ti)+c×log^{1+ϵ}(n) for some constant c. We also prove that the weak 2linkage problem remains NPcomplete if we require one of the two paths to be a shortest path while the other path has no restriction on the length.
DPcoloring is a relatively new coloring concept by Dvořák and Postle and was introduced as an extension of listcolorings of (undirected) graphs. It transforms the problem of finding a listcoloring of a given graph G with a listassignment L to finding an independent transversal in an auxiliary graph whose vertices are the pairs (v,c) with v in V(G) and c in L(v). In this paper, we extend the definition of DPcolorings to digraphs using the approach from NeumannLara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks’ type theorem regarding the DPchromatic number, which extends various results on the (list)chromatic number of digraphs.
This thesis studies combinatorial, algorithmic and complexity aspects of graph theory problems, and especially of problems related to the notions of walks, transitions and distances in graphs.
We first study the problem of traffic monitoring, in which we have to place as few censors as possible on the arcs of a graph to be able to retrace walks of objects. The characterization of instances of practical interests brings us to the notion of forbidden transitions, which strengthens the model of graphs. Our work on forbiddentransition graphs also includes the study of connecting transition sets, which can be seen as a translation to forbiddentransition graphs of the notion of spanning trees.
A large part of this thesis focuses on geometric graphs, which are graphs whose vertices are points of the real space and whose edges are determined by geometric distance between the vertices. This graphs are at the core of the famous HadwigerNelson problem and are of great help in our study of the density of sets avoiding distance 1 in various normed spaces. We develop new tools to study these problems and use them to prove the BachocRobins conjecture on several parallelohedra. We also investigate the case of the Euclidean plane and improve the bounds on the density of sets avoiding distance 1 and on its fractional chromatic number.
Finally, we study the complexity of graph homomorphism problems and establish dichotomy theorems for the complexity of locallyinjective homomorphisms to reflexive tournaments.
A subset A of R² is said to avoid distance 1 if for all x and y in A, xy≠1. In this paper we study the number m_1(R²) which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, m_1(R²) represents the highest proportion of the plane that can be filled by a set avoiding distance 1. This parameter is related to the fractional chromatic number χ_f(R²) of the plane.
We establish that m_1(R²) ≤ 0.25646 and χ_f(R²) ≥ 3.8992.
A forbidden transition graph is a graph defined together with a set of permitted transitions i.e. unordered pair of adjacent edges that one may use consecutively in a walk in the graph. In this paper, we look for the smallest set of transitions needed to be able to go from any vertex of the given graph to any other. We prove that this problem is NPhard and study approximation algorithms. We develop theoretical tools that help to study this problem.
For oriented graphs G and H, a homomorphism f: G → H is locallyinjective if, for every v in V(G), it is injective when restricted to some combination of the inneighbourhood and outneighbourhood of v. Two of the possible definitions of localinjectivity are examined. In each case it is shown that the associated homomorphism problem is NPcomplete when H is a reflexive tournament on three or more vertices with a loop at every vertex, and Polynomial when H is a reflexive tournament on two or fewer vertices.
The maximal density of a measurable subset of R^n avoiding Euclidean distance 1 is unknown except in the trivial case of dimension 1. In this paper, we consider the case of a distance associated to a polytope that tiles space, where it is likely that the sets avoiding distance 1 are of maximal density (1/2)^n, as conjectured by Bachoc and Robins. We prove that this is true for n=2, and for the Voronoï regions of the lattices A_n, n≥2.
This paper studies the problem of traffic monitoring which consists of differentiating a set of walks on a directed graph by placing sensors on as few arcs as possible. The problem of characterising a set of individuals by testing as few attributes as possible is already wellknown, but traffic monitoring presents new challenges that the previous models of separation fall short from modelling such as taking into account the multiplicity and order of the arcs in a walk. We introduce a new and stronger model of separation based on languages that generalises the traffic monitoring problem. We study three subproblems with practical applications and develop methods to solve them by combining integer linear programming, separating codes and language theory.
Despite most recent advances in structural variation discovery, many variants are yet to be discovered. Midsize insertions and deletions pose particularly involved algorithmic challenges. A prior method of ours (CLEVER) pointed out new ways. However, the resulting read alignment clusters tend to be too small and are heavily overlapping, which leads to losses in recall performance rates. Here we present a model based on \emph{weighted cluster editing}, which alleviates these issues: clusters are provably nonoverlapping and tend to be larger. In order to render the inherent optimization problem tractable on all, and not just discordant, read alignment data of a genome, we present a novel, principled heuristic, which solves the problem in time linear in the length of the genome. The heuristic is based on an exact polynomialtime algorithm for weighted cluster editing in onedimensional point graphs. We demonstrate that the new model improves recall rates achieved by CLEVER.